Quick Review: Quadrilaterals and Polygons

Quadrilateral

Types of Quadrilaterals:

• Parallelogram: In a parallelogram, opposite sides are parallel and congruent.
  Area = length * height
  Perimeter = 2 (length + breadth)
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• Rectangle: In a rectangle, opposite sides are parallel and congruent.
  Area = length * breadth
  Diagonal of the rectangle = √(l² + b²)
  Area = length * breadth
  Perimeter = 2 (length + breadth)
• Square: All sides and angles of a square are congruent.
  Length of the diagonal = l √2 (where l is the length of the side of a square)
  Area = (Side)²
  Perimeter = 4 X side
• Rhombus: All sides and all opposite angles of a rhombus are congruent. Also, the diagonals are perpendicular to and each other.
  Area = (a*b)/2 (where a and b are the lengths of the diagonals of a rhombus)
  Perimeter = 4 * length
• Trapezium: No sides, angles and diagonals of a trapezium are parallel to each other.
  Area = (1/2)h*(L+L₂)
  Perimeter = L + L₁ + L₂ + L₃
  

Important Points

• Of all the quadrilaterals of the same perimeter, the one with the maximum area is the square.
• The quadrilateral formed by joining the midpoints of the sides of any quadrilateral is always a parallelogram (rhombus in case of a rectangle, rectangle in case of a rhombus and square in case of a square).
• The quadrilateral formed by the angle bisectors of the angles of a parallelogram is a rectangle.
• For a rhombus □ABCD, if the diagonals are AC and BD, then AC² + BD² = 4 * AB².
• If a square is formed by joining the midpoints of a square, then the side of the smaller square = side of the bigger square/√2.
• If P is a point inside a rectangle □ABCD, then AP² + CP² = BP² + DP².
• The segment joining the midpoints of the non-parallel sides of a trapezium is parallel to the two parallel sides and is half the sum of the parallel sides.
• For a trapezium □ABCD, if the diagonals are AC and BD, and AB and CD are the parallel sides, then AC² + BD² = AD² + BC² + 2 * AB * BD.
• If the length of the sides of a cyclic quadrilaterals area, b, c and d, then it's area = √(s-a)(s-b)(s-c)(s-d) , where s is the semi - perimeter.
• The opposite angles of a cyclic quadrilateral are supplementary.

Polygons

Properties

• The sides a_n of regular inscribed polygons, where R is the radius of the circumscribed circle = a_n = 2R sin ^180o/n
• Area of a polygon of perimeter P and radius of in-circle r = (P*r)/2
• The sum of the interior angles of a convex POLYGON, having n sides is 180^o (n - 2).
• The sum of the exterior angles of a convex polygon, taken one at each vertex, is 360^o.
• The measure of an exterior angle of a regular n- sided polygon is 360^o/n .
• The measure of the interior angle of a regular n-sided polygon is (n-2)180^o/n
• The number of diagonals of in an n-sided polygon is n(n-3)/2